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A027199 Triangular array T read by rows: T(n,k) = number of partitions of n into an odd number of parts, each >=k.

%I A027199 #20 May 15 2023 11:14:05
%S A027199 1,1,1,2,1,1,2,1,1,1,4,1,1,1,1,5,2,1,1,1,1,8,2,1,1,1,1,1,10,3,1,1,1,1,
%T A027199 1,1,16,4,2,1,1,1,1,1,1,20,6,2,1,1,1,1,1,1,1,29,7,3,1,1,1,1,1,1,1,1,
%U A027199 37,10,4,2,1,1,1,1,1,1,1,1,52,12,5,2,1,1,1,1,1,1,1,1,1,66,17,6,3,1,1,1,1,1,1,1,1,1,1
%N A027199 Triangular array T read by rows: T(n,k) = number of partitions of n into an odd number of parts, each >=k.
%F A027199  T(n, k) = Sum{O(n, i)}, k<=i<=n, O given by A027185.
%F A027199 T(n,k) + A027200(n,k) = A026807(n,k). - _R. J. Mathar_, Oct 18 2019
%F A027199 G.f. of column k: x^k * Sum_{i>=0} x^(2*k*i)/Product_{j=1..2*i+1} (1-x^j). - _Seiichi Manyama_, May 15 2023
%e A027199 Triangle begins:
%e A027199    1;
%e A027199    1,  1;
%e A027199    2,  1, 1;
%e A027199    2,  1, 1, 1;
%e A027199    4,  1, 1, 1, 1;
%e A027199    5,  2, 1, 1, 1, 1;
%e A027199    8,  2, 1, 1, 1, 1, 1;
%e A027199   10,  3, 1, 1, 1, 1, 1, 1;
%e A027199   16,  4, 2, 1, 1, 1, 1, 1, 1;
%e A027199   20,  6, 2, 1, 1, 1, 1, 1, 1, 1;
%e A027199   29,  7, 3, 1, 1, 1, 1, 1, 1, 1, 1;
%e A027199   37, 10, 4, 2, 1, 1, 1, 1, 1, 1, 1, 1;
%e A027199   52, 12, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%o A027199 (PARI) T(n, k) = polcoef(x^k*sum(i=0, n, x^(2*k*i)/prod(j=1, 2*i+1, 1-x^j+x*O(x^n))), n); \\ _Seiichi Manyama_, May 15 2023
%Y A027199 Cf. A027185, A027193 (first column), A027194, A027195, A027196, A027197, A027198.
%K A027199 nonn,tabl
%O A027199 1,4
%A A027199 _Clark Kimberling_
%E A027199 More terms from _Seiichi Manyama_, May 15 2023